These ML schemes are trained offline, as a supervised learning problem, where the inputs \(\mathbf {x}_i\)  and outputs \(\mathbf {f}\)  are taken from a pre-computed dataset.
Because this training fails to account for feedbacks between \(\mathbf {g}_i\)  and \(\mathbf {f}\) , offline accuracy does not translate immediately into online accuracy when the ML is coupled to the fluid dynamics solver and used to simulate the weather or climate.
For example, offline training will often yield a numerically unstable scheme that causes an online simulation to crash within a matter of days [1]}, [2]}.
The reasons behind this are starting to emerge.
[2]} demonstrated with formal stability analysis that this instability is related to the linearized behavior of a neural network (NN) when coupled to idealized wave dynamics.
Also, feedback loops in coupled simulations can easily generate out-of-sample inputs unlike any in the training sample.
On the other hand, random forests (RFs) are more stable online, likely because they can only predict averages of observed samples [4]}, [5]}.
